An Italian dominating function (IDF) of a graph G is f : V(G) → {0, 1, 2} satisfying the condition that for every v ∈ V with f(v) = 0, Σu∈N(v)f(u) ≥ 2. The weight an IDF on sum f(V) Σv∈Vf(v) and domination number, γI (G), minimum IDF. perfect (PID) G, if vertex 0 total assigned by to neighbours exactly 2, i.e., all u are except one which 2 or two vertices w f(w) 1. PID-function Σu∈V(G) f(u). nu...

In a graph G=(V,E), where every vertex is assigned 0, 1 or 2, f an assignment such that 0 has at least one neighbor 2 and all vertices labeled by are independent, then called outer independent Roman dominating function (OIRDF). The domination strengthened if 1, 3, each two neighbors double (OIDRDF). weight of (OIDRDF) OIRDF the sum f(v) for v?V. (double) number (?oidR(G)) ?oiR(G) minimum taken ...

A subset S of the vertices of a graph G is an outer-connected dominating set, if S is a dominating set of G and G − S is connected. The outer-connected domination number of G, denoted by γ̃c(G), is the minimum cardinality of an OCDS of G. In this paper we generalize the outer-connected domination in graphs. Many of the known results and bounds of outer-connected domination number are immediate c...

A 2-outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of V (G)\D has a at least two neighbors in D, and the set V (G) \D is independent. The 2-outer-independent domination number of a graph G, denoted by γ 2 (G), is the minimum cardinality of a 2-outer-independent dominating set of G. We prove that for every nontrivial tree T of order n with l leav...

A total outer-independent dominating set of a graph G = (V (G), E(G)) is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V (G) \D is independent. The total outer-independent domination number of a graph G, denoted by γ t (G), is the minimum cardinality of a total outer-independent dominating set of G. We prove that for every tree T of order n ≥ 4, with l le...

A set D ⊆ V of a graph G = (V,E) is called an outer-connected dominating set of G if for all v ∈ V , |NG[v] ∩ D| ≥ 1, and the induced subgraph of G on V \D is connected. The Minimum Outer-connected Domination problem is to find an outer-connected dominating set of minimum cardinality of the input graph G. Given a positive integer k and a graph G = (V,E), the Outer-connected Domination Decision ...